CHANGS CONJECTURE FOR TRIPLES REVISITED MASAHIRO SHIOYA UNIVERSITY - - PDF document
CHANGS CONJECTURE FOR TRIPLES REVISITED MASAHIRO SHIOYA UNIVERSITY OF TSUKUBA 1 2 Chang type conjectures for pairs Suppose N = ( N ; R, ) is a structure for a countable first-order language with a distinguished unary pred- icate
CHANG’S CONJECTURE FOR TRIPLES REVISITED
MASAHIRO SHIOYA UNIVERSITY OF TSUKUBA
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Chang type conjectures for pairs Suppose N = (N; R, · · · ) is a structure for a countable first-order language with a distinguished unary pred- icate symbol interpreted by R ⊂ N. N has type (ν, ν′) if |N| = ν and |R| = ν′. (ν, ν′) ։ (µ, µ′) iff ∀N of type (ν, ν′) ∃M of type (µ, µ′) s.t. M ≺ N. Originally Chang conjectured (ω2, ω1) ։ (ω1, ω).
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Consistency of Chang’s original conjecture Theorem (Silver). Con(an ω1-Erd˝
Con((ω2, ω1) ։ (ω1, ω)).
λ (say) to ω2 by the Silver collapse S(ω1, λ).
MA can be removed and the Levy collapse works as well. (Shelah)
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Chang’s conjecture for triples
We consider structures with two distinguished unary predicates. (ω3, ω2, ω1) ։ (ω2, ω1, ω) implies (ω3, ω2) ։ (ω2, ω1). PFA implies that the Levy collapse forces (ω3, ω2) ̸։ (ω2, ω1). (Foreman–Magidor)
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Consistency of Chang’s conjecture for pairs Theorem (Kunen). Con(a huge cardinal exists) implies Con((ω3, ω2) ։ (ω2, ω1)). κ is huge with target λ iff ∃j : V → M s.t. κ = crit(j), λ = j(κ), λM ⊂ M.
S(κ, λ) ֒ → j(P). The final model is given by P ∗ ˙ S(κ, λ).
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Consistency of Chang’s conjecture for triples Theorem (Foreman). Con(a 2-huge cardinal exists) implies Con((ω3, ω2, ω1) ։ (ω2, ω1, ω)). κ is 2-huge iff ∃j : V → M s.t. κ = crit(j), j2(κ)M ⊂ M.
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Open problem It has been open for 30 years whether Con((ω4, ω3, ω2, ω1) ։ (ω3, ω2, ω1, ω)). Perhaps Con(3-huge) would suffice. But how?
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A new model of Chang’s conjecture for pairs
µ
∏
β∈[µ,κ)∩R
S(β, κ) ∗ ˙ S(κ, λ) forces κ = µ+, λ = µ++ and (µ++, µ+) ։ (µ+, µ).
µ
∏ stands for the < µ-support product. R denotes the class of regular cardinals.
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From embeddings to projections Let P, R be posets. A map π : P → R is a projection if: (1) π is order-preserving, (2) π(1P ) = 1R, (3) r′ ≤R π(p) → ∃p∗ ≤P p s.t. π(p∗) ≤R r′. If π : P → R be a projection, then we get e : R ֒ → B(P) by r → ∑ {p ∈ P : π(p) ≤ r}. Conversely e : R ֒ → P gives rise to a projection : P → B(R).
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Term spaces Suppose ˙ S is a P-name for a poset. The term space is the “set” T(P, ˙ S) = { ˙ s : ˙ s is a P-name ∧ ˙ s ∈ ˙ S}
s′ ≤ ˙ s iff ˙ s′ ˙ ≤ ˙ s. As sets, P × T(P, ˙ S) = P ∗ ˙ S.
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Basic lemma of term spaces Lemma (Laver). The identity map id : P × T(P, ˙ S) → P ∗ ˙ S is a projection. Using the lemma we will get a projection j(P) → P ∗ ˙ S(κ, λ), where P =
µ
∏
β∈[µ,κ)∩R
S(β, κ).
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The Silver collapse (with slight modification) Suppose κ < λ are regular cardinals with λ inaccessible. The Silver collapse S(κ, λ) is the set of s : δ × d → λ such that
A cardinal γ is κ-closed if γ<κ = γ. S(κ, λ) has nice properties of the original Silver collapse.
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Identifying the term space Main Lemma. Suppose P has κ-cc and size ≤ κ. Then S(κ, λ) is isomorphic to a dense subset of T(P, ˙ S(κ, λ)).
following form: id × i : P × S(κ, λ) → P ∗ ˙ S(κ, λ). Results should hold for suitable modifications of other canonical collapses as well.
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Proof sketch The dense set is D = { ˙ s : ∃δ < κ∃d ⊂ [κ, λ) dom ˙ s = δ × d}. Define i : s ∈ S(κ, λ) → ˙ s ∈ D by dom ˙ s = dom s ∧ ˙ s(α, γ) = τ(s(α, γ)). Here P-names τ(ξ) are chosen so that for every κ-closed γ {τ(ξ) : ξ < γ} is a 1-1 enumeration of all P-names ˙ α s.t. ˙ α < γ.
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Master conditions (Extending elementary embeddings) Suppose
A condition p∗ ∈ j(P) is a master condition for (j and) ϕ if ∀¯ p ≤ p∗ ¯ p ≤ j(ϕ(¯ p)). If ¯ G ⊂ j(P) is generic and contains a master condition for ϕ, then (j ◦ ϕ)“ ¯ G ⊂ ¯ G and j can be extended to j : V [ϕ[ ¯ G]] → M[ ¯ G] in V [ ¯ G]. (ϕ[ ¯ G] = the filter over P generated by ϕ“ ¯ G.)
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Getting master conditions Lemma (Kunen). Suppose
S(κ, λ) is a projection,
Then there is a master condition (1j(P ), ˙ s∗) for π+ : j(P ∗ ˙ S(κ, λ)) → P ∗ ˙ S(κ, λ).
X by ˙ X = {(j( ˙ s), ¯ p) : π(¯ p) ≤ (1P , ˙ s)}, and let ˙ s∗ = ∪ ˙ X.
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Proof for a new model Let j : V → M witness that κ is huge with target λ. Let P =
µ
∏
β∈[µ,κ)∩R
S(β, κ). We claim that P ∗ ˙ S(κ, λ) works. Define a projection π : j(P) → P ∗ ˙ S(κ, λ) by j(P) =
µ
∏
β∈[µ,λ)
S(β, λ)
∼
− − − − →
µ
∏
β∈[µ,κ)
S(β, λ) ×
µ
∏
β∈[κ,λ)
S(β, λ) (Q rsκ)×prκ (
µ
∏
β∈[µ,κ)
S(β, κ) ) × S(κ, λ) id × i (
µ
∏
β∈[µ,κ)
S(β, κ) ) ∗ ˙ S(κ, λ). By Kunen’s lemma there is a master condition for π+ : j(P ∗ ˙ S(κ, λ)) → P ∗ ˙ S(κ, λ) (1j(P ), ˙ s∗), below which j can be extended to j : V P ∗ ˙
S(κ,λ) → M j(P ∗ ˙ S(κ,λ)).
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Toward Chang’s conjecture for triples Theorem (Foreman). Suppose that κ is 2-huge. Let µ < κ be regular. Then κ = µ+ and (µ+++, µ++, µ+) ։ (µ++, µ+, µ) in some forcing extension.
We claim that P(κ) ∗ ˙ Q(κ, λ) ∗ ˙ S(λ, θ)
P(λ) ∗ ˙ Q(λ, θ) ∗ ˙ S(θ, j(θ)). Claim 1. There is a projection: P(λ) → P(κ) ∗ ˙ Q(κ, λ). Claim 2. P(λ) forces that there is a projection: ˙ Q(λ, θ)) → ˙ S(λ, θ)P (κ)∗ ˙
Q(κ,λ).
Claim 3. There is a master condition for the projection: j(P(κ) ∗ ˙ Q(κ, λ)) → P(κ) ∗ ˙ Q(κ, λ).
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Generalization 1 Suppose
S is an R-name for a poset. Let P ⋆π ˙ S (or P ⋆ ˙ S) be the set P × T(R, ˙ S)
s′) ≤ (p, ˙ s) iff p′ ≤P p ∧ π(p′) R ˙ s′ ˙ ≤ ˙ s. If P = R and π = id, P ⋆ ˙ S = P ∗ ˙ S as posets.
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Laver type lemma 1
S is an R-name for a poset. Then id : P × T(R, ˙ S) → P ⋆π ˙ S is a projection.
Then there is a projection of the following form: id × i : P × S(κ, λ) → P ⋆ ˙ S(κ, λ)R.
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Generalization 2 Suppose X, Y are disjoint sets of ordinals, and for β ∈ X ∪ Y
Sβ is an Rβ-name for a poset. Let P ⋆
κ
∏
β∈X
˙ Sβ ×
E
∏
β∈Y
˙ Sβ be the set P ×
κ
∏
β∈X
T(Rβ, ˙ Sβ) ×
E
∏
β∈Y
T(Rβ, ˙ Sβ)
p′ ≤P p ∧ dom q′ ⊃ dom q ∧ ∀β ∈ dom q πβ(p′) β q′(β) ˙ ≤β q(β).
E
∏ stands for the Easton support product. β denotes the forcing relation w.r.t. Rβ.
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Laver type lemma 2 Lemma. id : P ×
κ
∏
β∈X
T(Rβ, ˙ Sβ) ×
E
∏
β∈Y
T(Rβ, ˙ Sβ) → P ⋆
κ
∏
β∈X
˙ Sβ ×
E
∏
β∈Y
˙ Sβ is a projection.
Then there is a projection of the form id × ∏
β∈X∪Y
iβ from P ×
κ
∏
β∈X
S(κβ, λ) ×
E
∏
β∈Y
S(κβ, λ) to P ⋆
κ
∏
β∈X
˙ S(κβ, λ)Rβ ×
E
∏
β∈Y
˙ S(κβ, λ)Rβ .
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Main forcing 1 Let M be the class of Mahlo cardinals together with µ. For κ ∈ M with κ > µ define P(κ) =
E
∏
β∈[µ,κ)∩R
β
∏
α∈[µ,β]∩M
S(β, κ) . P(κ) is µ-closed and a subset of Vκ.
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Main forcing 2-1 By recursion on κ ∈ M we define for all pairs of κ < λ from M
κ : P(λ) → R(κ, λ).
First let R(µ, λ) = P(λ) and πλ
µ = id.
Suppose R(α, κ) and πκ
α have been defined for α ∈ [µ, κ) ∩ M.
Define R(κ, λ) = P(κ) ⋆
κ
∏
α∈[µ,κ]∩M
˙ S(κ, λ)R(α,κ) ×
E
∏
β∈(κ,λ)∩R
˙ S(β, λ) , where we stipulate R(κ, κ) = P(κ) and πκ
κ = id.
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Main forcing 2-2 Let πλ
κ be the composition of the following projections:
P(λ)
ψλ
κ
− − − − → P(κ) × ( κ ∏
α∈[µ,κ]∩M S(κ, λ) × E
∏
β∈(κ,λ)∩R S(β, λ)
)
ϕλ
κ
− − − − → R(κ, λ). Here ϕλ
κ is the projection defined from Main lemma.
To define ψλ
κ, first identify
P(λ) =
E
∏
β∈[µ,λ)∩R
β
∏
α∈[µ,β]∩M
S(β, λ) with
E
∏
β∈[µ,κ)∩R
β
∏
α∈[µ,β]∩M
S(β, λ) ×
κ
∏
α∈[µ,κ]∩M
S(κ, λ) ×
E
∏
β∈(κ,λ)∩R
β
∏
α∈[µ,β]∩M
S(β, λ) . Now let ψλ
κ be the map (identified with)
∏
β∈[µ,κ)∩R
∏
α∈[µ,β]∩M
(rsκ |S(β, λ)) × id × ∏
β∈(κ,λ)∩R
prκ |
β
∏
α∈[µ,β]∩M
S(β, λ) .
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Remark It might seem more natural to define P(κ) =
E
∏
β∈[µ,κ)∩R
β
∏
α∈[µ,β]∩M
S(β, κ) ×
E
∏
γ∈(β,κ)∩R
S(γ, κ) , and ψλ
κ : P(λ) → P(κ) ×
κ
∏
α∈[µ,κ]∩M
S(κ, λ) ×
E
∏
γ∈(κ,λ)∩R
S(γ, λ) by identifying P(λ) =
E
∏
β∈[µ,λ)∩R
β
∏
α∈[µ,β]∩M
S(β, λ) ×
E
∏
γ∈(β,λ)∩R
S(γ, λ) with
E
∏
β∈[µ,κ)∩R
β
∏
α∈[µ,β]∩M
S(β, λ) ×
E
∏
γ∈(β,λ)∩R
S(γ, λ) ×
E
∏
β∈[κ,λ)∩R
β
∏
α∈[µ,β]∩M
S(β, λ) ×
E
∏
γ∈(β,λ)∩R
S(γ, λ) . This would not work for some reason to be discussed later.
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Notation In what follows let ˙ Q(κ, λ) =
κ
∏
α∈[µ,κ]∩M
˙ S(κ, λ)R(α,κ) ×
E
∏
β∈(κ,λ)∩R
˙ S(β, λ), so that R(κ, λ) = P(κ) ⋆ ˙ Q(κ, λ). Also let ¯ Q(κ, λ) =
κ
∏
α∈[µ,κ]∩M
S(κ, λ) ×
E
∏
β∈(κ,λ)∩R
S(β, λ), so that we have P(λ)
ψλ
κ
− − − − → P(κ) × ¯ Q(κ, λ)
ϕλ
κ
− − − − → P(κ) ⋆ ˙ Q(κ, λ).
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Getting master conditions 1 (Kunen type lemma) It remains to get a master condition for the projection: j(P(κ) ⋆ ˙ Q(κ, λ)) → P(κ) ⋆ ˙ Q(κ, λ). Recall ˙ Q(κ, λ) =
κ
∏
α∈[µ,κ]∩M
˙ S(κ, λ)P (α)⋆ ˙
Q(α,κ) × E
∏
β∈(κ,λ)∩R
˙ S(β, λ).
E
∏
β∈X ˙
S(β, λ) is a projection,
Then there is a master condition (1j(P ), q∗) for π+ : j(P ⋆
E
∏
β∈X
˙ S(β, λ)) → P ⋆
E
∏
β∈X
˙ S(β, λ).
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Getting master conditions 2 (Foreman’s lemma) Lemma (Foreman). Suppose
S is an R-name for a poset,
j(P)
j(π)
− − − − → j(R)
ϕP
ϕR P ⋆ ˙ S − − − − →
π×id
R ∗ ˙ S
P : j(P) → P,
s∗) is a master conditions for ϕ+
R : j(R ∗ ˙
S) → R ∗ ˙ S. Then (1j(P ), ˙ s∗) is a master condition for ϕ+
P : j(P ⋆ ˙
S) → P ⋆ ˙ S.
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Foreman’s diagram 1 To get a master condition for the projection j(P(κ) ⋆ ˙ S(κ, λ)P (γ)⋆ ˙
Q(γ,κ)) → P(κ) ⋆ ˙
S(κ, λ)P (γ)⋆ ˙
Q(γ,κ).
the following diagram suffices: j(P(κ)) − − − − → j(P(γ) ⋆ ˙ Q(γ, κ))
S(κ, λ)P (γ)⋆ ˙
Q(γ,κ) −
− − − → (P(γ) ⋆ ˙ Q(γ, κ)) ∗ ˙ S(κ, λ).
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Foreman’s diagram 2
Then the following diagram commutes: P(λ)
πλ
γ
− − − − → P(γ) ⋆ ˙ Q(γ, λ)
πλ
κ
id × ˙
ρ
P(κ) ⋆ ˙ Q(κ, λ) P(γ) ⋆ ( ˙ Q(γ, κ) × ˙ S(κ, λ) )
id × prγ
id ×i∗ P(κ) ⋆ ˙ S(κ, λ)P (γ)⋆ ˙
Q(γ,κ) −
− − − →
πκ
γ ×id
( P(γ) ⋆ ˙ Q(γ, κ) ) ∗ ˙ S(κ, λ). ˙ ρ ≅ ∏
α∈[µ,γ]∩M
( ˙ rsκ| ˙ S(γ, λ)P (α)⋆ ˙
Q(α,γ))
× ∏
β∈(γ,κ)∩R
( ˙ rsκ| ˙ S(β, λ) ) × prκ, where ˙ Q(γ, λ) =
κ
∏
α∈[µ,γ]∩M
˙ S(γ, λ)P (α)⋆ ˙
Q(α,γ) × E
∏
β∈(γ,λ)∩R
˙ S(β, λ) ≅
κ
∏
α∈[µ,γ]∩M
˙ S(γ, λ)P (α)⋆ ˙
Q(α,γ) ×
E
∏
β∈(γ,κ)∩R
˙ S(β, λ) ×
E
∏
β∈[κ,λ)∩R
˙ S(β, λ) .
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Auxiliary diagram 1 P(λ)
ψλ
κ
Q(κ, λ)
ϕλ
κ
− − − − → P(κ) ⋆ ˙ Q(κ, λ)
id × prγ
id × prγ P(κ) × S(κ, λ)
id ×iγ
− − − − → P(κ) ⋆ ˙ S(κ, λ)P (γ)⋆ ˙
Q(γ,κ) ψκ
γ ×id
πκ
γ ×id
( P(γ) × ¯ Q(γ, κ) ) × S(κ, λ)
ϕκ
γ×iγ
− − − − → ( P(γ) ⋆ ˙ Q(γ, κ) ) ∗ ˙ S(κ, λ).
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Auxiliary diagram 2 P(λ)
ψλ
γ
Q(γ, λ)
ϕλ
γ
− − − − → P(γ) ⋆ ˙ Q(γ, λ)
id ׯ ρ
id × ˙
ρ
P(γ) × ( ¯ Q(γ, κ) × S(κ, λ) ) P(γ) ⋆ ( ˙ Q(γ, κ) × ˙ S(κ, λ) )
ϕκ
γ×id
id ( P(γ) ⋆ ˙ Q(γ, κ) ) × S(κ, λ)
id ×i
− − − − → ( P(γ) ⋆ ˙ Q(γ, κ) ) ⋆ ˙ S(κ, λ)P (γ)
id
id ×i∗ ( P(γ) ⋆ ˙ Q(γ, κ) ) × S(κ, λ)
id ×iγ
− − − − → ( P(γ) ⋆ ˙ Q(γ, κ) ) ∗ ˙ S(κ, λ).